Invar36 alloy is a low expansion alloy, and the triply periodic minimal surfaces (TPMS) structures have excellent lightweight, high energy absorption capacity and superior thermal and acoustic insulation properties. It is, however, difficult to manufacture by traditional processing methods. Laser powder bed fusion (LPBF) as a metal additive manufacturing technology, is extremely advantageous for forming complex lattice structures. In this study, five different TPMS cell structures, Gyroid (G), Diamond (D), Schwarz-P (P), Lidinoid (L), and Neovius (N) with Invar36 alloy as the material, were prepared using the LPBF process. The deformation behavior, mechanical properties, and energy absorption efficiency of these structures under different load directions were studied, and the effects and mechanisms of structure design, wall thickness, and load direction were further investigated. The results show that except for the P cell structure, which collapsed layer by layer, the other four TPMS cell structures all exhibited uniform plastic collapse. The G and D cell structures had excellent mechanical properties, and the energy absorption efficiency could reach more than 80%. In addition, it was found that the wall thickness could adjust the apparent density, relative platform stress, relative stiffness, energy absorption, energy absorption efficiency, and deformation behavior of the structure. Printed TPMS cell structures have better mechanical properties in the horizontal direction due to intrinsic printing process and structural design.